Evaluations of some series of the type $\sum_{k=0}^\infty(ak+b)x^k/\binom{mk}{nk}$ (2204.08275v6)

Abstract: In this paper, via the beta function we evaluate some series of the type $\sum_{k=0}^\infty(ak+b)x^k/\binom{mk}{nk}$. We completely determine the values of $$\sum_{k=1}^\infty\frac {k^rx^k}{\binom{3k}k}\ \left(-\frac{27}4<x<\frac{27}4\right)\ \text{and}\ \sum_{k=1}^\infty\frac {k^rx^k}{\binom{4k}{2k}}\ (-16<x<16)$$ for $r=0,\pm1$. For example, we prove that $$\sum_{k=0}^\infty\frac{(49k+1)8^k}{3^k\binom{3k}k}=81+16\sqrt3\,\pi \ \ \text{and}\ \ \sum_{k=0}^\infty\frac{10k-1}{\binom{4k}{2k}}=\frac{4\sqrt 3}{27}\pi.$$ We also establish the following efficient formula for computing $\log n$ with $1<n\le 85/4$: \begin{align*} &\sum_{k=0}^\infty\frac{(2(n^2+6n+1)^2(n^2-10n+1)k+P(n))(n-1)^{4k}} {(-n)^k(n+1)^{2k}\binom{4k}{2k}}\ \ \ &=6n(n+1)(n-1)^3\log n-32n(n+1)^2(n^2-4n+1), \end{align*} where $$P(n):=n^6-58n^5+159n^4+52n^3+159n^2-58n+1.$$